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  "body": "\n## Summary\n\nConsider a singular integral operator $a \\varphi + b S \\varphi + \\int_\\Gamma k \\varphi$ on a closed curve $\\Gamma$.\nIts index is the winding number around $0$ of the function $(a - b)/(a + b)$ on $\\Gamma$.\nIt is an integer computed from the coefficients alone, and it does not depend on the regular part $k$.\nBy the [[Noether theorems|FDC1BEB8FW]], it equals the number of independent solutions of the homogeneous equation minus that of the adjoint homogeneous equation.\nSo it is the [[Fredholm index|2V9Q4KHY1T]] of the operator.\nIt decides how far the classical Fredholm alternative fails for the equation.\n\n## Explanations\n\n### L1 human\n\nFor a matrix equation, the number of free parameters in the solutions minus the number of conditions on the data is a fixed number, the difference of the dimensions.\nFor a singular integral equation, both numbers are finite, and their difference is the index.\nThe surprise is that the index can be read off from a picture.\nFollow the complex number $(a(t) - b(t))/(a(t) + b(t))$ as $t$ runs around the curve, and count how many times it turns around $0$.\nFor example, on the unit circle with $a(t) = (t + 1)/2$ and $b(t) = (t - 1)/2$, the quotient is $1/t$, which turns once clockwise, so the index is $-1$.\n",
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